I-need-help said:
Okay, thank you. I understand that, I'm just not sure how to get a ratio from that, with no other information... I'm not even sure if I need to work out a ratio actually. Oh well, thanks anyway.
Firstly, your original answer 2:3 was correct - but your uncertainty indicated you were not sure why.
When doing ratios, I just use the formulas and do a grand divide to produce the ratio.
In this case we want the ratio of masses.
Well I know density is mass/ volume [ σ = M/V] so
M = Vσ
Now the ratio: firstly put subscripts on the variable - I would use c for copper and a for aluminium
M
a = V
aσ
a
and
M
c = V
cσ
c
In ratio form:
M
a/M
c = V
a/V
c x σ
a/σ
c
We thus know
M
a/M
c = V
a/V
c x 1/3
since we were given the ratio of the densities.
So now we need the ratio of the Volumes to complete this.
Each wire is effectively a cylinder
V =πr
2h
For the wire, h = length of the wire - which is the same for both wires - so the ratio reduces to.
V
a/V
c = r
a2/r
c2
So now we need the ratio of Radii [or diameters?]
Resistance is given by"
R = ρL/A
Since Area here is that of the circular wire,
R = ρL/∏r
2
transposing
r
2 = ρL/∏R
This gives
r
a2 = ρ
aL
a/∏
aR
a
and
r
c2 = ρ
cL
c/∏
cR
c
Now for these wires, Length and resistance [and of course ∏] are the same, so the ratio simplifies to
r
a2/r
c2 = ρ
a/ρ
c
Substituting back into:
V
a/V
c = r
a2/r
c2
gives
V
a/V
c = ρ
a/ρ
c
Then back into:
M
a/M
c = V
a/V
c x 1/3
gives
M
a/M
c = ρ
a/ρ
c x 1/3
which gives
M
a/M
c = 2 x 1/3
which is 2/3 or 2:3 if you like.
While this has been lengthy to type out, when written it is much quicker.
Note: Normally the line:
r
a2/r
c2 = ρ
a/ρ
c
would be expressed as
r
a/r
c = √[ρ
a/ρ
c]
but I knew my previous formula had r
a2/r
c2 in it so I left it as was.
You can use this ratio technique to find the ratio of anything:
eg Ratio of two accelerations
F = ma → a = F/m
so
a
1 = F
1/m
1
and
a
2 = F
2/m
2
a
1/a
2 = F
1/F
2 x m
2/m
1
[Note that since m was in the denominator, is appears "upside down" as a ratio.]
SO once we know the ratio of the forces, and the ratio of the masses we can work out the ratio of the accelerations, without calculating/knowing the actual value of each acceleration.